I have a question about regressing a covariate on the slope. If the variable on slope is significant, does that mean that the slope itself is significant as well as the variable being a predictor of the slope? Can you recommend an article related to this topic?

In the regression of the slope growth factor on a covariate, a signficant relationship says that the covariate has a significant influence on the slope growth factor. This implies the slope growth factor has variance. See the Raudenbush and Bryk book on multilevel modeling.

I have a follow up question if you don't mind. I have a model with a single freely estimated time point. The slope is significantly predicted by a covariate, however the mean for the quadratic for that class is also significant. I know that typically one can't interpret a linear slope when a quadratic trend is observed, however having freely estimated a time point, I've rendered the quadratic meaningless right? Should I then only attend to the slope? Also is the Raudenbush and Bryk book you recommend "Hierarchical Linear Models: Applications and Data Analysis Methods"? I ask because I always end up buying the wrong book by the right authors.